ThinkstockMail Flow
When you first set up Office 365, you are prompted to configure your domains' DNS to work with Office 365. Microsoft provides records for mail routing (MX), autodiscover (CNAME), and SPF (Sender Protection Framework). Failure to apply the correct settings here can mean complete loss of mail flow or lack of client connectivity.
List all authorized domains, including third-party services, as authoritative domains in Office 365's Exchange admin center to ensure email delivery to all of your recipients.
SPF is a special consideration. This record type is used to inform other mail systems whether email from your domain is coming from an authorized system. The record provided by Microsoft is suitable if the only place your email will ever originate is Office 365. Often this is not the case, however, because you might use third-party tools such as Salesforce or MailChimp to send email on behalf of your domain or apps. In order to ensure delivery to your recipients, be sure to include any of these services in your SPF record. More information on SPF syntax can be found at The SPF Project.
Once you have full access to the Exchange admin center, you should verify that all of your domain names are listed and declared as authoritative (or of the appropriate relay type as necessary) under Mail Flow > Accepted Domains, as shown in the screenshot above.